Every day, Alvin commutes from Here to There. Some days, the roads are clear while other days they are quite busy. The state of the road (i.e., "clear" or "busy") on any given day is independent of that on any other day. On any day, the length of time it takes Alvin to get to There from Here is equal to the time spent at red lights plus a random component, both of which are independent of one another. The number of red lights he encounters is a geometric random variable with mean five. On clear days, he spends one minute at each light. On days where there is traffic, Alvin spends two minutes at each light. The random component, X, also depends on the state of the road. Specifically on clear days,

while on busy days

fx|c(x|C) = { e ̄x/2u(x)

fx|c(x|B) == e-x/^u(x)

where u(x) is the unit step function. Finally, it is known that on average Alvin spends 13 minutes getting from Here to There.

1. What is the expected time and variance of Alvin's travel time on a clear day?

2. What is the probability that on a given day the roads are clear?

3. On a clear day, what is the probability that it takes Alvin less than 3 minutes to get to work given that he hits one red light?

4. On a busy day, what is the probability mass function for the number of red lights Alvin hits given that his travel time is less than five minutes?

5. Suppose you wanted to solve a hypothesis-testing problem for the state of the roads given a single observation of Alvin's commute time. For what range of times can you guarantee zero error?

6. Suppose you are given both the length of time for Alvin's commute as well as the number of red lights he hit. What is the maximum likelihood rule for the hypothesis-testing problem of determining whether the roads were clear or busy that day?